All Questions

Consider the polynomial set: $$ f_{ij} = (t_i - t_j)x_i x_j + x_i - x_j, \quad (1 \leq j < i \leq 2n) $$ where $ t_1, t_2, \dots, t_{2n} $ are pairwise distinct. Let's look at the non-crossing ...

Let $F_n^{(k)}(x)= \sum_j {\binom{n+(k-1)j}{kj} x^j}$ and $G_n^{(k)}(x)= \sum_j {\binom{n+j}{kj} x^j}.$ I am interested in the coefficients ${a_{n,k,j}}$ such that $$G_n^{(k)}(x)=\sum_{j\geq0 }{a_{n,...

The sequence of polynomials $$P_n=\sum_{k=0}^{\lfloor(2n-1)/3\rfloor} \frac{(2n-2k-1)!(2n-2k-2)!}{k!(n-k)!(n-k-1)!(2n-3k-1)!}x^k$$ satisfies apparently the identities $$0=\sum_{j=0}^nP_{n-j}(P_j-(-x)^...

Define $N(F)$ to be the number of monomials of a multi-variable polynomial $F$. For example $N(x^2y+3xy-y^5)=3$. If $\mathbf{x}=(x_1,\dots,x_n)$ and $F_n(\mathbf{x})=\prod_{k=1}^n(x_1+\cdots+x_k)$ ...

Given a multi-variable polynomial $F$, denote the number of monomials by $N(F)$. Take for instance, \begin{align*}N(x(x+y)+(x+y)^2-(x-y)^2)=N(x^2+5xy)&=2 \qquad \text{and} \\ N((x+z)(x+y)^2)=N(x^3 ...

Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta < \pi$, there are natural constructions which associate to this data: a noncrossing matching on $2n$ ...